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Proper fractions

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					      Proper fractions
The value of the numerator is less
than the value of the denominator.

Proper in this case does not mean
correct or best.
         Improper fractions
• The value of the numerator is greater than
  or equal to the value of the denominator.
               Mixed numbers
• Meaning of

    2
  5
    3
    Writing mixed numbers as
        improper fractions

• The algorithm that is taught in schools
  obscures the meaning. This is true for
  many algorithms, which are “efficient”
  ways of carrying out operations.
Write mixed number as improper
     fraction and vice versa

      1
  2   3



 23
 4
     Operations with fractions
• Addition

• Subtraction

• Multiplication

• Division
Adding and subtracting fractions
1/2 + 1/3
        Multiplying fractions
• Repeated addition model



• Area model
     Multiplication of fractions
• Fraction as operator




• The multiplication algorithm is best
  explained by the area model.
2/3 of 2 1/2
Mixed number times mixed
        number
           Dividing fractions
• Division of fractions is most easily
  understood as repeated subtraction.



   12 2
11 divided by 1 1/2
        Multiplicative Inverses
• We know that division is the inverse of
  multiplication.


        10  2  5
               1
        10      5
               2
        Multiplicative inverses
• The multiplicative        1
  inverse of a is 1/a     a 1
                            a


                          a b
• The multiplicative        1
  inverse of a/b is b/a   b a
          Dividing fractions
• Because division is the inverse operation
  of multiplication, dividing a number by a
  fraction is equivalent to multiplying the
  number by the multiplicative inverse,
  called the reciprocal, of the fraction.
           Exploration 5.12
• “Drawn to scale”
• Part 1 Use reasoning not algorithms to
  answer #1
• Part 2 Choose a model from the list that
  was not represented in the problems and
  make up a story problem using the fraction
  ¾. Are there any models that are not
  possible with fractions? Explain.
     Operations with fractions
• Addition




• Subtraction
     Operations with fractions
• Multiplication
     Operations with fractions
• Division
            Exploration 5.13
• Begin in class and finish for homework: 5.13
  Part 1: #2-7 Part 2: Choose one of the models
  from the list that was not illustrated in the
  problems in Part 1 and write a story problem
  using the fraction ¾. Also, are there any models
  that are not possible with fractions? Explain.
• Homework problems from the textbook:
  pp. 303-305: 3b,d,e,f, 13, 21, 22, 25
  Note that in #3, you should not use algorithms to
  calculate the result; use reasoning to decide the
  answer to the question.
                Extra Practice
• 1. You have from 10:00 - 11:30 to do a project. At 11,
  what fraction of time remains? At 11:20, what fraction of
  time remains?
• Use a diagram to explain how you know. Are there
  certain diagrams that are more effective? Discuss this
  with your group.
            Extra Practice
• 2. Is 10/13 closer to 1/2 or 1?
• Use a diagram to explain how you know.
  Are there certain diagrams that are more
  effective? Discuss this with your group.
             Extra Practice
• 3. If a/b = 3/4, will the value of
  (a + x)/(b + x) be less than, equal to, or
  greater than 3/4.
• Use a diagram to explain how you know.
  Are there certain diagrams that are more
  effective? Discuss this with your group.
             Exploration 5.14
•   Read the directions carefully and do #1
•   Discuss with your partner
•   Do #2
•   Discuss with your partner
•   Do # 3
   Homework for Wednesday
• Exploration 5.14
• Read section 5.3 in your textbook
• Do problems pp. 305-307: 30, 31, 33, 36,
  41, 44a,c,d,h,k, 45a, 48

				
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